Fixed-rank matrix factorizations and Riemannian low-rank optimization
This provides a geometric optimization framework for machine learning tasks involving fixed-rank matrices, such as linear regression, but it is incremental as it generalizes prior work on symmetric matrices.
The paper tackles the problem of optimizing smooth cost functions over fixed-rank matrices by adopting Riemannian quotient manifolds, leading to gradient descent and trust-region algorithms that scale to high-dimensional problems and compete with state-of-the-art methods in numerical experiments.
Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix.